Find the smallest positive integer that is a multiple of 35 whose digits are all the same.

To be divisible by 35, it should be divisible by 5.
So, the last digit must be 0 or 5. As 0 is not possible,
the integer must have all 5s.
555..n times should be divisible by 7.
{5*(1111..)} mod 7=0
=> (111..n times) mod 7=0
=> 10^n-1 mod 7=0
=> 3^n mod 7=1
Order of 3 modulo 7 is 6.
So, n=6 => 111111 is divisible by 7 => 555555 is the smallest
integer divisible by 35.

Posted by Praneeth on 2007-08-24 08:02:09